Fredholmness vs. Spectral Discreteness for first-order differential operators
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- by N. Anghel PDF
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Abstract:
It is shown that for essentially self-adjoint first-order differential operators $D$, acting on sections of bundles over complete (non-compact) manifolds, Fredholmness vs.$\!$ Spectral Discreteness is the same as ‘$\exists c>0$, $D$ is $c$-invertible at infinity’ vs. ‘$\forall c>0$, $D$ is $c$-invertible at infinity’. An application involving the spectral theory of electromagnetic Dirac operators is then given.References
- Nicolae Anghel, An abstract index theorem on noncompact Riemannian manifolds, Houston J. Math. 19 (1993), no. 2, 223–237. MR 1225459
- Paul R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis 12 (1973), 401–414. MR 0369890, DOI 10.1016/0022-1236(73)90003-7
- M. Gromov and B. Lawson, Positive Scalar Curvature and the Dirac Operator on Complete Riemannian Manifolds, Publ. Math. IHES, $\mathbf {58}$, 295-408, (1983).
- Akira Iwatsuka, Magnetic Schrödinger operators with compact resolvent, J. Math. Kyoto Univ. 26 (1986), no. 3, 357–374. MR 857223, DOI 10.1215/kjm/1250520872
- A. M. Molčanov, On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order, Trudy Moskov. Mat. Obšč. 2 (1953), 169–199 (Russian). MR 0057422
- Naohiro Suzuki, Discrete spectrum of electromagnetic Dirac operators, Proc. Amer. Math. Soc. 128 (2000), no. 3, 819–825. MR 1628440, DOI 10.1090/S0002-9939-99-05073-X
- E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford, at the Clarendon Press, 1946 (German). MR 0019765
- Hermann Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), no. 2, 220–269 (German). MR 1511560, DOI 10.1007/BF01474161
Additional Information
- N. Anghel
- Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
- MR Author ID: 26280
- Email: anghel@unt.edu
- Received by editor(s): November 19, 2014
- Received by editor(s) in revised form: January 22, 2015
- Published electronically: June 26, 2015
- Communicated by: Varghese Mathai
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 693-701
- MSC (2010): Primary 35P05, 58J50; Secondary 81Q10, 81V10
- DOI: https://doi.org/10.1090/proc12741
- MathSciNet review: 3430845